Investments, tenth edition

  The Index Portfolio as an Investment Asset 

 The process of charting the efficient frontier using the single-index model can be pursued 

much like the procedure we used in Chapter 7, where we used the Markowitz model to find 

the optimal risky portfolio. Here, however, we can benefit from the simplification the index 

model offers for deriving the input list. Moreover, portfolio optimization highlights another 

advantage of the single-index model, namely, a simple and intuitively revealing representation 

of the optimal risky portfolio. Before we get into the mechanics of optimization in this setting, 

however, we start by considering the role of the index portfolio in the optimal portfolio. 

 Suppose the prospectus of an investment company limits the universe of investable 

assets to only stocks included in the S&P 500 portfolio. In this case, the S&P 500 index 

captures the impact of the economy on the large stocks the firm may include in its portfolio. 

Suppose that the resources of the company allow coverage of only a relatively small subset 

of this so-called  investable universe.  If these analyzed firms are the only ones allowed in 

the portfolio, the portfolio manager may well be worried about limited diversification. 

 A simple way to avoid inadequate diversification is to include the S&P 500 portfolio as 

one of the assets of the portfolio. Examination of Equations 8.8 and 8.9 reveals that if we 

treat the S&P 500 portfolio as the market index, it will have a beta of 1.0 (its sensitivity to 

itself), no firm-specific risk, and an alpha of zero—there is no nonmarket component in its 

expected return. Equation 8.10 shows that the covariance of any security,  i,  with the index 

is     b

i

s

M

2

.  To distinguish the S&P 500 from the  n  securities covered by the firm, we will 

designate it the ( n   1  1)th asset. We can think of the S&P 500 as a  passive portfolio  that the 

manager would select in the absence of security analysis. It gives broad market  exposure 

without the need for expensive security analysis. However, if the manager is willing to 

bod61671_ch08_256-290.indd   272

bod61671_ch08_256-290.indd   272

6/21/13   4:10 PM

6/21/13   4:10 PM

Final PDF to printer

---

  C H A P T E R  

8

 Index 

Models 

273

engage in such research, she may devise an  active portfolio  that can be mixed with the 

index to provide an even better risk–return trade-off.